Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps
arXiv:1003.5592
Abstract
We consider C^2 families of C^4 unimodal maps f_t whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure of f_t depends differentiably on t, as a distribution of order 1. The proof uses transfer operators on towers whose level boundaries are mollified via smooth cutoff functions, in order to avoid artificial discontinuities. We give a new representation of the acim for a Benedicks-Carleson map f_t, in terms of a single smooth function and the inverse branches of f_t along the postcritical orbit. Along the way, we prove that the twisted cohomological equation v(x)=α(f (x)) - f'(x) α(x) has a continuous solution α, if f is Benedicks-Carleson and v is horizontal for f. In v3 we added a note containing 3 comments regarding minor typos.
v2: Typos corrected. Banach spaces (Prop 4.10, Prop 4.11, Lem 4.12, Appendix B, Section 6) cleaned up: H^1_1 Sobolev space replaces C^1 and BV, L^1 replaces C^0, and H^2_1 replaces C^2. Details added (e.g. Remark 4.9). The map f_0 is now C^4. 61 pages. v3: added 3 comments written in 2013 about minor typos